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State the Remainder Theorem.

April 6, 2026Mathematics

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Answer

\text{-3}

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To find the remainder when $x^3 - 5x - 7$ is divided by $x+1$ , we can use the Remainder Theorem.

State the Remainder Theorem.
The Remainder Theorem states that if a polynomial $P(x)$ is divided by $x-c$ , the remainder is $P(c)$ .
Identify the polynomial $P(x)$ and the value of $c$ .
The given polynomial is $P(x) = x^3 - 5x - 7$ .
The divisor is $x+1$ . To find $c$ , we set the divisor to zero:
$x+1 = 0$
$x = -1$
So, $c = -1$ .
Substitute $c = -1$ into the polynomial $P(x)$ .
The remainder is $P(-1)$ .
$P(-1) = (-1)^3 - 5(-1) - 7$
Calculate the value.
$P(-1) = -1 - (-5) - 7$ $P(-1) = -1 + 5 - 7$ $P(-1) = 4 - 7$ $P(-1) = -3$
The remainder is $\text{-3}$ .

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